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We prove that one can define the relation ‖, with ab ‖ cd to be read as ‘a = b or c = d or ab and cd are parallel lines (or coincide)’ positively existentially in Lω1ω1 in terms of 6= and the ternary relation B of betweenness, with B(abc) to be read as ‘b lies between a and c’ in Archimedean ordered affine geometry. We also show that a self-map of an Archimedean ordered translation plane or of a flat affine plane which preserves both B and ¬B must be a surjective affine mapping. Mathematics Subject Classification: 51G05, 51F20, 51F05.

Positive existential definability of parallelism in terms of betweenness in Archimedean ordered affine geometry